Talk:Beth number

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[edit] Difference between aleph numbers and beth numbers

I don't see any difference between anAleph number and a Bet number. Can they be used interchangeably? Then why have both terms? 65.74.59.37 08:41, 21 January 2006 (UTC)Chani

They are different if (and only if) GCH fails. It's not usual to assume GCH, so they can't usually be used interchangeably. --Zundark 11:53, 21 January 2006 (UTC)

[edit] Gimel numbers

What's gimel used for? Is it used interchangeably with beth? Because they look similar? Awmorp 23:26, 7 June 2006 (UTC)

I have no idea. I've never seen it outside of Wikipedia. — Arthur Rubin | (talk) 23:59, 7 June 2006 (UTC)
I think that gimel numbers are defined analogously to aleph numbers and beth numbers, but with \gimel_{\alpha+1}=\gimel_\alpha^{\mathrm{cf}(\gimel_\alpha)}. However, I'm not aware of any published reference for this, so I've removed the mention of them from the article (which in any case was confusing due to the lack of explanation). --Zundark 08:21, 8 June 2006 (UTC)

[edit] Merging Beth2 into this article

Any discussion is taking place at Talk:Beth_two#Merge_into_Beth_Numbers. LambiamTalk 17:54, 2 April 2006 (UTC)

  • Merge completed (yesterday), as best I could. If I missed anything relevant that should have been merged, the last version before merge is still there. — Arthur Rubin | (talk) 14:43, 25 July 2006 (UTC)

[edit] Clarification needed

"sup" is not defined or linked in any way, as in:

\beth_{\lambda}=\sup\{ \beth_{\alpha}:\alpha<\lambda \}.

It would be helpful if it was; I can't be the only reader who finds this complete double-Dutch. --Michael C. Price talk 03:07, 14 February 2007 (UTC)

If the axiom of choice holds, then these cardinals can be identified with their initial ordinals. In that case, the supremum is the least ordinal greater than all those in the set. If the axiom of choice does not hold, then the "obvious" generalization is the direct limit of the sequence of representative sets. So if A is the initial set, then
\beth_{\lambda} (|A|) = | \bigcup_{\alpha < \lambda} B_{\alpha} |
where
\beth_{\alpha} (|A|) = | B_{\alpha} |
and the Bs are nested. JRSpriggs 08:44, 14 February 2007 (UTC)
Even if the axiom of choice holds, the direct limit of cardinals is not always defined, I'm afraid. The direct limit for a strictly increasing set of cardinals works, which applies here, but the limit of ω1 (the ordinal corresponding to \aleph_1) \aleph_0's can be \aleph_0 or \aleph_1. The cardinality of the union of the specific nested set of sets makes sense, but "direct limit" of cardinals, in general, does not. — Arthur Rubin | (talk) 15:45, 14 February 2007 (UTC)
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